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Unattainable speed of light and (anti)gravity...

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ocalhoun
The reason you can't accelerate anything to the speed of light is because the faster it goes, the heavier (and harder to accelerate) it gets, right?

What if the accelerating force is gravity? The difference there is that the heavier something gets, the more it wants to accelerate.

I guess it might be impossible to attain a speed greater than the speed of light towards a gravity well, because the object would hit the center of the gravity well before getting to that speed.

For the sake of eliminating that objection, I'm assuming a huge source of 'antigravity'. Something that instead of drawing things towards it, repels them away, and which has a HUGE amount of this force. It also works the same way in that it repels heavy objects more than it does light objects.

I know the answer is probably that this wouldn't allow an object to break the 'light barrier', but my question is: why not?
Indi
Assuming GR is correct, and an observer in the same frame of reference as the "anti-gravity well", i'd guess: because the further you travel from the source of anti-gravity, the weaker its push will get (the inverse square law), so object will keep accelerating, but not at a constant rate. It will accelerate more slowly the faster it gets. The curve will be asymptotic with a the asymptote as c.

i can't prove this without solving the equations, though. What you'd do is get the push force with:

Use F = ma to figure out the acceleration of the object (taking into account relativistic mass, for m) and using a = dv/dt, solve for v.

Eh-heh, good luck. ^_^;
ocalhoun
Indi wrote:


Eh-heh, good luck. ^_^;

But, does that take into account that the force applied to the object will increase as the mass increases due to speed?
Indi
Yes, that what the relativistic mass () is.
ocalhoun
You know, I'm actually curious enough to tackle these equations... They're complex, but don't have anything in them I can't handle.

Let me make sure I have the correct assumptions of what each variable represents.

So, the whole thing assembled would be:

(Gravitational Constant ( ( mass of object 1 * mass of object 2) / ?r?^2))= (mass / sqrt(1-((velocity^2)/(speed of light^2))) * Acceleration

That leaves me with two questions...

1: what does this r stand for?
2: getting max speed from acceleration...

...

Oh, wait, I may have figured it out.
While the acceleration due to gravity may be unlimited, the acceleration required for speed of light is infinite. Though the force of gravity could increase without limit, the absolute value of it will never be infinite. So, it may accelerate really close to c, but still can't reach it, right?
Indi
ocalhoun wrote:
You know, I'm actually curious enough to tackle these equations... They're complex, but don't have anything in them I can't handle.

^_^; Bold!

ocalhoun wrote:
Let me make sure I have the correct assumptions of what each variable represents.

So, the whole thing assembled would be:

(Gravitational Constant ( ( mass of object 1 * mass of object 2) / ?r?^2))= (mass / sqrt(1-((velocity^2)/(speed of light^2))) * Acceleration

That leaves me with two questions...

1: what does this r stand for?
2: getting max speed from acceleration...

It's so hard to properly express equations with just basic BBCode, so i put something together that you can read through. It will require a browser that supports SVG and MathML, but Firefox comes with both built in.

That last nasty equation is the one you have to to solve. It's a differential equation. i didn't bother to solve it, but i offered a set of solutions that are probably the solution (or something very, very similar). They show how position just flies off to infinity (the object never stops moving), but both velocity and acceleration are limited. Velocity approaches c, and acceleration shrinks down to virtually zero. That should help it make more physical sense.

But to answer your questions:
  1. r is the position - in this case, the distance between the two objects.
  2. To get the max speed from the acceleration, you just find out where acceleration is zero. In this case (using the sample solutions), acceleration only reaches zero at time infinity (in other words, never). At infinity, the hyperbolic tangent is 1, so the maximum velocity, reached only after infinite time, is c.


ocalhoun wrote:
Oh, wait, I may have figured it out.
While the acceleration due to gravity may be unlimited, the acceleration required for speed of light is infinite. Though the force of gravity could increase without limit, the absolute value of it will never be infinite. So, it may accelerate really close to c, but still can't reach it, right?

i'm not sure i understand what you mean. Acceleration is unlimited in that it never gets to zero, but it is limited in that it's the largest when the two objects are closest, and it gets smaller the farther apart they get.

You don't need infinite acceleration to achieve the speed of light - constant acceleration would do the trick. If you could accelerate at 2 m/s for long enough, you'd break the speed of light. The problem is that the acceleration provided by your antigravity source is not constant, it follows the inverse square law.

As you can see from the equations, the size of the antigravity force doesn't matter. Set K to any value you like... you'll see the solution is the same: the only thing that changes is the rate that you approach c, but you still never get there or cross it. (This is because the hyperbolic tangent function tanh(x) is between 0 and 1 for all values of x greater than 0 - and if you want to extend to negative values, tanh(x) is between -1 and 1 for all values between positive and negative infinity. So with c*tanh(K t), no matter what K or t is, your speed is limited to + or - c.)

(FYI, i suspect that K = G * M₀. i'm basing this and my suggested solutions based on experience with these kinds of equations, but i might be way off. G is constant, but set M₀ to any positive nonzero value (because a mass can only be positive and not zero), and v = c * tanh(G * M₀ * t) will range from 0 at t=0 to c at t=∞. The only thing that M₀ changes is how quickly it approaches c.)
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